Multiply by $u_t$, integrate and see what happens.
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Michael RenardyNov 29 '12 at 2:28

You will likely to get a good answer if you ask this at MathStackExchange.
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timurNov 29 '12 at 3:57

1

It's much the same as how the ODE $\dfrac{d^2 y}{dt^2} + a \dfrac{dy}{dt} + b y$ for $a,b > 0$ represents a damped oscillator. In fact, if you take $u(x,t) = u(t) X(x)$ where $X$ is an eigenfunction of the Laplacian, this is exactly what you get.
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Robert IsraelNov 29 '12 at 6:29